I am learning about relativity and am not quite sure how to think of spacetime. From a mathematical perspective, spacetime is a manifold i.e. a topological space for which about any point there exists an open neighbourhood that is homeomorphic to an open subset of R^4.
However a topological space (and thus manifold) is fundamnetally, in a mathematician's definition, a set of elements, X with some additional structure. Namely, that certain elements of the power set of X belong to a set- the topology on X- which we say are 'the open subsets of X'.
Pure math out of the way, I am not sure how to think of the points in spacetime. I have heard many physicists refer to the elements of the speactime manifold as 'events'. Yet this does not seem correct to me. Generally- and in most usage other than when you explicitly ask a physicist what spacetime is made of- and 'event' is an 'occurance', like "spaceship emitting a light pulse", "spaceship absorbing a light pulse" etc. It seems that events are ascribed to a specific spacetime point. More rigorously, an 'event' is uniquely assigned to an element of the manifold.
The whole point of the topological view is that the spacetime manifold has an existence independent of anything occuring in it. But I am unsure whether my understanding of 'events' and 'spacetime elements' is correct. And if so, I still find it somewhat unsatisfactory and would like to know what a phyiscist regards as the nature of the spacetime points in which specific events don't occur. These points have an existence as spacetime ppoints to which an event could have been ascribed (I laugh at my use of the past tense!) but wasn't. What is the ontology of these elements from a physicist's perspective?